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In [[thermodynamics]] and [[thermal physics]], the '''Gouy-Stodola theorem''' is an important theorem for the quantification of irreversibilities in an open [[Thermodynamic system|system]], and aids in the [[exergy]] analysis of [[Thermodynamic process|thermodynamic processes]]. It asserts that the rate at which [[Work (thermodynamics)|work]] is lost during a process, or at which exergy is destroyed, is proportional to the rate at which [[entropy]] is generated, and that the proportionality coefficient is the [[temperature]] of the ambient heat reservoir.<ref name=":3">{{Cite book|last=Bejan|first=Adrian|url=https://www.worldcat.org/oclc/32204119|title=Entropy generation minimization : the method of thermodynamic optimization of finite-size systems and finite-time processes|date=1996|publisher=CRC Press|isbn=0-8493-9651-4|location=Boca Raton|oclc=32204119}}</ref> In the literature, the theorem often appears in a slightly modified form, changing the proportionality coefficient.<ref name=":0">{{Cite journal|last=Pal|first=Rajinder|date=2017-03-30|title=On the Gouy–Stodola theorem of thermodynamics for open systems|url=http://journals.sagepub.com/doi/10.1177/0306419017697413|journal=International Journal of Mechanical Engineering Education|language=en|volume=45|issue=2|pages=194–206|doi=10.1177/0306419017697413|s2cid=125782232|issn=0306-4190}}</ref>

The theorem is named jointly after the French physicist [[Louis Georges Gouy|Georges Gouy]] and Slovak physicist [[Aurel Stodola]], who demonstrated the theorem in 1889 and 1905 respectively.<ref name=":5">{{Cite journal|last=Lucia|first=Umberto|date=2014-09-01|title=The Gouy-Stodola Theorem in Bioenergetic Analysis of Living Systems (Irreversibility in Bioenergetics of Living Systems)|journal=Energies|language=en|volume=7|issue=9|pages=5717–5739|doi=10.3390/en7095717|issn=1996-1073|doi-access=free}}</ref><ref name=":6">{{Cite journal|last=Lucia|first=Umberto|date=2013-03-01|title=Stationary open systems: A brief review on contemporary theories on irreversibility|url=https://linkinghub.elsevier.com/retrieve/pii/S0378437112009922|journal=Physica A: Statistical Mechanics and Its Applications|language=en|volume=392|issue=5|pages=1051–1062|doi=10.1016/j.physa.2012.11.027|bibcode=2013PhyA..392.1051L}}</ref> Gouy used it while working on exergy and utilisable energy, and Stodola while working on steam and gas engines.<ref>{{Cite journal|last=Gouy|date=1889|title=Sur l'énergie utilisable|url=http://www.edpsciences.org/10.1051/jphystap:018890080050101|journal=Journal de Physique Théorique et Appliquée|volume=8|issue=1|pages=501–518|doi=10.1051/jphystap:018890080050101|issn=0368-3893}}</ref><ref>G. Gouy, Sur les transformation et l’équilibre en Thermodynamique, Comptes Rendus de l’Acadèmie des Sciences Paris 108 (10) (1889) 507–509</ref><ref>G. Gouy, Sur l’énergie utilisable et le potentiel thermodynamique, Note de M. Gouy, Comptes Rendus de l’Acadàmie des Sciences Paris 108 (10) (1889) 794.</ref><ref>{{Cite book|last=Aurel|first=Stodola|title=Steam Turbine|publisher=D. Van Nostrand Co.|year=1905|isbn=978-1297500220|location=New York, NY, USA}}</ref>

== Overview ==
The Gouy-Stodola theorem is often applied upon an open thermodynamic system, which can exchange heat with some [[Thermal reservoir|thermal reservoirs]]. It holds both for systems which cannot exchange mass, and systems which mass can enter and leave.<ref name=":0" /><ref name=":2">{{Cite journal|last=Pal|first=Rajinder|date=2017-02-10|title=Demystification of the Gouy-Stodola theorem of thermodynamics for closed systems|url=http://journals.sagepub.com/doi/10.1177/0306419016689501|journal=International Journal of Mechanical Engineering Education|language=en|volume=45|issue=2|pages=142–153|doi=10.1177/0306419016689501|s2cid=125522221|issn=0306-4190}}</ref>
[[File:A sketch of a system, in thermal contact with multiple reservoirs, with mass flowing in and out, producing work..jpg|alt=A system along with reservoirs|thumb|400x400px|A system along with reservoirs]]
Observe such a system, as sketched in the image shown, as it is going through some [[Thermodynamic process|process]]. It is in contact with multiple reservoirs, of which one, that at temperature <math>T_0</math>, is the environment reservoir. During the process, the system produces work and generates entropy. Under these conditions, the theorem has two general forms.

=== Work form ===
The reversible work is the maximal useful work which can be obtained, <math>W_{rev}=W_{max}</math>, and can only be fully utilized in an ideal [[Reversible process (thermodynamics)|reversible process]]. An irreversible process produces some work <math>W_{actual}</math>, which is less than <math>W_{rev}</math>. The lost work is then <math>W_{lost}=W_{rev}-W_{actual}</math>; in other words, <math>W_{lost}</math> is the work which was lost or not exploited during the process due to irreversibilities.<ref name=":0" /><ref name=":1">{{Cite journal|last=Lucia|first=Umberto|date=2016-04-27|title=The Gouy-Stodola theorem as a variational principle for open systems|url=https://doi.org/10.1478/AAPP.941A4|journal=Atti della Accademia Perloritana dei Pericolanti. Classe di Scienze Fisiche, Matematiche e Naturali|volume=94|issue=1|pages=A4–1–A4-16|doi=10.1478/AAPP.941A4|s2cid=115131389}}</ref> In terms of lost work, the theorem generally states<math display="block">\dot{W}_{lost}=T_0\dot{S_{g}}</math>where <math>\dot{W}_{lost}</math> is the rate at which work is lost, and <math>\dot{S_{g}}</math> is the rate at which entropy is generated. Time derivatives are denoted by dots. The theorem, as stated above, holds only for the entire thermodynamic universe - the system along with its surroundings, together:<math display="block">\dot{W}_{lost,tot}=T_0\dot{S}_{g,tot}</math>where the index "tot" denotes the total quantities produced within or by the entire universe.

Note that <math>\dot{W}_{lost}</math> is a relative quantity, in that it is measured in relation to a specific thermal reservoir. In the above equations, <math>\dot{W}_{lost}</math> is defined in reference to the environment reservoir, at <math>T_0</math>. When comparing the actual process to an ideal, reversible process between the same endpoints (in order to evaluate <math>W_{rev}</math>, so as to find the value of <math>W_{lost}</math>), only the heat interaction with the reference reservoir <math>T_0</math> is allowed to vary. The heat interactions between the system and other reservoirs are kept the same. So, if a different reference reservoir <math>T_{ref}</math> is chosen, the theorem would read <math>\dot{W}_{lost,tot}=T_{ref}\dot{S}_{g,tot}</math>, where this time <math>\dot{W}_{lost}</math> is in relation to <math>T_{ref}</math>, and in the corresponding reversible process, only the heat interaction with <math>T_{ref}</math> is different.<ref name=":0" /> 

By integrating over the lifetime of the process, the theorem can also be expressed in terms of final quantities, rather than rates: <math>{W}_{lost,tot}=T_0{S}_{g,tot}</math>.<ref name=":1" />

==== Adiabatic case ====
The theorem also holds for [[Adiabatic process|adiabatic processes]]. That is, for closed systems, which are not in thermal contact with any heat reservoirs.

Similarly to the non-adiabatic case, the lost work is measured relative to some reference reservoir <math>T_0</math>. Even though the process itself is adiabatic, the corresponding reversible process may not be, and might require heat exchange with the reference reservoir. Thus, this can be thought of as a special case of the above statement of the theorem - an adiabatic process is one for which the heat interactions with all reservoirs are zero, and in the reversible process, only the heat interaction with the reference thermal reservoir may be different.<ref name=":0" /><ref name=":2" />

The adiabatic case of the theorem holds also for the other formulation of the theorem, presented below. 

=== Exergy form ===
The [[exergy]] of the system is the maximal amount of useful work that the system can generate, during a process which brings it to equilibrium with its environment, or the amount of energy available. During an [[irreversible process]], such as heat exchanges with reservoirs, exergy is destroyed. Generally, the theorem states that<math display="block">\dot{\psi}_{d}=T_0\dot{S_{g}}</math>where <math>\dot{\psi}_{d}</math> is the rate at which exergy is destroyed, and <math>\dot{S_{g}}</math> is the rate at which entropy is generated.<ref name=":0" /><ref name=":2" /> As above, time derivatives are denoted by dots. 

Unlike the lost work formulation, this version of the theorem holds for both the system (the control volume) and for its surroundings (the environment and the thermal reservoirs) separately:<math display="block">\dot{\psi}_{d,sys}=T_0\dot{S}_{g,sys}</math>and<math display="block">\dot{\psi}_{d,surr}=T_0\dot{S}_{g,surr}</math>where the index "sys" denotes quantities produced within or by the system itself, and "surr" within or by the surroundings. Therefore, summing these two forms, the theorem also holds for the thermodynamic universe as a whole:<math display="block">\dot{\psi}_{d,tot}=\dot{\psi}_{d,sys}+\dot{\psi}_{d,surr}=T_0\dot{S}_{g,sys}+T_0\dot{S}_{g,surr}=T_0\dot{S}_{g,tot}</math>where the index "tot" denotes the total quantities of the entire universe.

Thus, the exergy formulation of the theorem is less limited, as it can be applied on different regions separately. Nevertheless, the work form is used more often.

The proof of the theorem, in both forms, uses the [[first law of thermodynamics]], writing out the terms <math>\dot{W}_{lost}</math>, <math>\dot{\psi}_{d}</math>, and <math>\dot{S_{g}}</math> in the relevant regions, and comparing them.

== Modified coefficient and effective temperature ==
In many cases, it is preferable to use a slightly modified version of the Gouy-Stodola theorem in work form, where <math>T_0</math> is replaced by some effective temperature. When this is done, it often enlarges the scope of the theorem, and adapts it to be applicable to more systems or situations. For example, the corrections elaborated below are only necessary when the system exchanges heat with more than one reservoir - if it exchanges heat only at the environmental temperature <math>T_0</math>, the simple form above holds true.<ref name=":4">{{Cite journal|last1=Holmberg|first1=Henrik|last2=Ruohonen|first2=Pekka|last3=Ahtila|first3=Pekka|date=2009-10-30|title=Determination of the Real Loss of Power for a Condensing and a Backpressure Turbine by Means of Second Law Analysis|journal=Entropy|language=en|volume=11|issue=4|pages=702–712|doi=10.3390/e11040702|bibcode=2009Entrp..11..702H|issn=1099-4300|doi-access=free}}</ref> Additionally, modifications may change the reversible process to which the real process is compared in calculating <math>\dot{W}_{lost}</math>. 

The modified theorem then reads<math display="block">\dot{W}_{lost}=T_{eff}\dot{S_{g}}</math>where <math>T_{eff}</math> is the effective temperature. 

For a flow process, let <math>s_{1}</math> denote the specific entropy (entropy per unit mass) at the inlet, where mass flows in, and <math>s_{2}</math> the specific entropy at the outlet, where mass flows out. Similarly, denote the specific [[Enthalpy|enthalpies]] by <math>h_{1}</math> and <math>h_{2}</math>. The inlet and outlet, in this case, function as initial and final states a process: mass enters the system at an initial state (the inlet, indexed "1"), undergoes some process, and then leaves at a final state (the outlet, indexed "2").

This process is then compared to a reversible process, with the same initial state, but with a (possibly) different final state. The theoretical specific entropy and enthalpy after this ideal, isentropic process are given by <math>s_{2,rev}</math> and <math>h_{2,rev}</math>, respectively. When the actual process is compared to this theoretical reversible process and <math>\dot{W}_{lost}</math> is evaluated, the proper effective temperature is given by<math display="block">T_{eff}=\frac{h_{2}-h_{2,rev}}{s_{2}-s_{2,rev}}</math>In general, <math>T_{eff}</math> lies somewhere in between the final temperature in the actual process <math>T_{2}</math> and the final temperature in the theoretical reversible process <math>T_{2,rev}</math>.<ref name=":3" /><ref name=":0" /><ref name=":4" /> 

This equation above can sometimes be simplified. If both the [[pressure]] and the [[specific heat capacity]] remain constant, then the changes in enthalpy and entropy can be written in terms of the temperatures, and<ref name=":0" /><ref name=":4" /><ref name=":7">{{Cite journal|last1=Sachdeva|first1=Gulshan|last2=Bilash|first2=Ram|date=2015-04-01|title=Thermodynamic Analysis Of A Vapor Absorption System Using Modified Gouy-Stodola Equation|url=https://zenodo.org/record/1100478|language=en|doi=10.5281/ZENODO.1100478}}</ref><ref name=":8">{{Cite journal|last1=Jain|first1=Vaibhav|last2=Sachdeva|first2=Gulshan|last3=Kachhwaha|first3=S.S.|date=2015-01-01|title=Thermodynamic modelling and parametric study of a low temperature vapour compression-absorption system based on modified Gouy-Stodola equation|url=https://linkinghub.elsevier.com/retrieve/pii/S0360544214012821|journal=Energy|language=en|volume=79|pages=407–418|doi=10.1016/j.energy.2014.11.027}}</ref><math display="block">T_{eff}=\frac{T_{2}-T_{2,rev}}{\ln T_{2}-\ln T_{2,rev}}=\frac{T_{2}-T_{2,rev}}{\ln\frac{T_{2}}{T_{2,rev}}}</math>However, it is important to note that this version of the theorem doesn't relate the exact values which the original theorem does. Specifically, in comparing the actual process to a reversible one, the modified version allows the final state to be different between the two. This is in contrast to the original version, wherein reversible process is constructed to match so that the final states are the same.<ref name=":0" /><ref name=":4" />

== Applications ==
In general, the Gouy-Stodola theorem is used to quantify irreversibilities in a system and to perform exergy analysis. That is, it allows one to take a thermodynamic system and better understand how [[Inefficiency|inefficient]] it is (energy-wise), how much work is lost, how much room there is for improvement and where. The [[second law of thermodynamics]] states, in essence, that the entropy of a system only increases. Over time, thermodynamic systems tend to gain entropy and lose energy (in approaching equilibrium): thus, the entropy is "somehow" related to how much exergy or potential for useful work a system has. The Gouy-Stodola theorem provides a concrete link. For the most part, this is how the theorem is used - to find and quantify inefficiencies in a system. 

=== Flow processes ===
{{Main|Flow process}}
A [[flow process]] is a type of [[thermodynamic process]], where matter flows in and out of an open system called the [[control volume]]. Such a process may be [[Steady state|steady]], meaning that the matter and energy flowing into and out of the system are constant through time. It can also be unsteady, or transient, meaning that the flows may change and differ at different times.

Many proofs of the theorem demonstrate it specifically for flow systems. Thus, the theorem is particularly useful in performing exergy analysis on such systems.<ref name=":0" /><ref>{{Cite journal|last1=Lampinen|first1=Markku J|last2=Wikstén|first2=Ralf|date=2006-01-01|title=Theory of Effective Heat-Absorbing and Heat-Emitting Temperatures in Entropy and Exergy Analysis with Applications to Flow Systems and Combustion Processes|url=https://www.degruyter.com/document/doi/10.1515/JNETDY.2006.012/html|journal=Journal of Non-Equilibrium Thermodynamics|volume=31|issue=3|page=257|doi=10.1515/JNETDY.2006.012|bibcode=2006JNET...31..257L|s2cid=123707485|issn=0340-0204}}</ref><ref>{{Cite journal|last=Pal|first=Rajinder|date=2018-04-01|title=Quantification of irreversibilities in practical cyclic processes using exergy analysis and Gouy-Stodola theorem|url=https://doi.org/10.1177/0306419017720427|journal=International Journal of Mechanical Engineering Education|language=en|volume=46|issue=2|pages=118–137|doi=10.1177/0306419017720427|s2cid=125161762|issn=0306-4190}}</ref>

=== Vapor compression and absorption ===
{{See also|Heat pump and refrigeration cycle|Refrigeration#Cyclic refrigeration|Vapor-compression refrigeration}}
The Gouy-Stodola theorem is often applied to [[Heat pump and refrigeration cycle|refrigeration cycles]]. These are thermodynamic cycles or mechanical systems where external work can be used to move heat from low temperature sources to high temperature sinks, or vice versa. Specifically, the theorem is useful in analyzing [[Vapor-compression refrigeration|vapor compression]] and [[Absorption refrigerator|vapor absorption]] refrigeration cycles. 

The theorem can help identify which components of a system have major irreversibilities, and how much exergy they destroy. It can be used to find at which temperatures the performance is optimal, or what size system should be constructed. Overall, that is, the Gouy-Stodola theorem is a tool to find and quantify inefficiencies in a system, and can point to how to minimize them - this is the goal of exergy analysis. When the theorem is used for these purposes, it is usually applied in its modified form.<ref name=":4" /><ref name=":7" /><ref name=":8" /><ref>{{Cite journal|last1=Sachdeva|first1=Gulshan|last2=Jain|first2=Vaibhav|last3=Kachhwaha|first3=S. S.|date=2013-11-25|title=EXERGY ANALYSIS OF A VAPOR COMPRESSION–VAPOR ABSORPTION CASCADE SYSTEM|url=https://www.worldscientific.com/doi/abs/10.1142/S2010132513500260|journal=International Journal of Air-Conditioning and Refrigeration|language=en|volume=21|issue=4|pages=1350026|doi=10.1142/S2010132513500260|issn=2010-1325}}</ref><ref>{{Cite journal|last1=Jain|first1=Vaibhav|last2=Sachdeva|first2=Gulshan|last3=Kachhwaha|first3=Surendra S.|date=2016-12-15|title=Estimation of Optimum Heat Exchanger Area of Vapor Compression-absorption Integrated System Using Modified Irreversibility Approach|url=https://linkinghub.elsevier.com/retrieve/pii/S1876610216313807|journal=Energy Procedia|language=en|volume=90|pages=57–68|doi=10.1016/j.egypro.2016.11.170|doi-access=free}}</ref>

=== In ecology ===
Macroscopically, the theorem may be useful environmentally, in ecophysics. An [[ecosystem]] is a complex system, where many factors and components interact, some biotic and some abiotic. The Gouy-Stodola theorem can find how much entropy is generated by each part of the system, or how much work is lost. Where there is human interference in an ecosystem, whether the ecosystem continues to exist or is lost may depend on how many irreversibilities it can support. The amount of entropy which is generated or the amount of work the system can perform may vary. Hence, two different states (for example, a healthy forest versus one which has undergone significant deforestation) of the same ecosystem may be compared in terms of entropy generation, and this may be used to evaluate the sustainability of the ecosystem under human interference.<ref>{{Cite journal|last1=Diaz-Mendez|first1=S.E.|last2=Sierra-Grajeda|first2=J.M.T.|last3=Hernandez-Guerrero|first3=A.|last4=Rodriguez-Lelis|first4=J.M.|date=2013-11-01|title=Entropy generation as an environmental impact indicator and a sample application to freshwater ecosystems eutrophication|url=https://linkinghub.elsevier.com/retrieve/pii/S0360544213007937|journal=Energy|language=en|volume=61|pages=234–239|doi=10.1016/j.energy.2013.09.042}}</ref><ref>{{Cite journal|last1=Díaz-Méndez|first1=Sósimo E.|last2=Rodríguez-Lelis|first2=José María|last3=Hernández-Guerrero|first3=Abel|date=2011-08-31|title=A general exergy-based environmental impact index|url=http://link.springer.com/10.1007/s12206-011-0219-0|journal=Journal of Mechanical Science and Technology|language=en|volume=25|issue=8|pages=1979–1985|doi=10.1007/s12206-011-0219-0|s2cid=59483293|issn=1738-494X}}</ref>

=== In biology ===
The theorem is also useful on a more microscopic scale, in [[biology]]. Living systems, such as [[Cell (biology)|cells]], can be analyzed thermodynamically. They are rather complex systems, where many energy transformations occur, and they often waste heat. Hence, the Gouy-Stodola theorem may be useful, in certain situations, to perform exergy analysis on such systems. In particular, it may help to highlight differences between healthy and diseased cells. 

Generally, the theorem may find applications in fields of [[biomedicine]], or where biology and physics cross over, such as [[Biochemical engineering|biochemical engineering thermodynamics]].<ref name=":5" /><ref>{{Cite journal|last=Lucia|first=Umberto|date=2016-11-15|title=Econophysics and bio-chemical engineering thermodynamics: The exergetic analysis of a municipality|url=https://linkinghub.elsevier.com/retrieve/pii/S0378437116303946|journal=Physica A: Statistical Mechanics and Its Applications|language=en|volume=462|pages=421–430|doi=10.1016/j.physa.2016.06.119|bibcode=2016PhyA..462..421L}}</ref>

=== As a variational principle ===
A [[variational principle]] in physics, such as the [[Principle of Least Action|principle of least action]] or [[Fermat's principle]] in optics, allows one to describe the system in a global manner and to solve it using the [[calculus of variations]]. In thermodynamics, such a principle would allow a [[Lagrangian mechanics|Lagrangian]] formulation. The Gouy-Stodola theorem can be used as the basis for such a variational principle, in thermodynamics. It has been proven to satisfy the necessary conditions.<ref name=":5" /><ref name=":6" /><ref name=":1" />

This is fundamentally different from most of the theorem's other uses - here, it isn't being applied in order to locate components with irreversibilities or loss of exergy, but rather helps give some more general information about the system.

== References ==
{{Reflist}}

[[Category:Thermodynamics]]

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